1600edo

← 1599edo

1600edo

1601edo →

Prime factorization	26 × 52
Step size	0.75 ¢
Fifth	936\1600 (702 ¢) (→ 117\200)
Semitones (A1:m2)	152:120 (114 ¢ : 90 ¢)
Consistency limit	37
Distinct consistency limit	37

The 1600 equal divisions of the octave (1600edo), or the 1600-tone equal temperament (1600tet), 1600 equal temperament (1600et) when viewed from a regular temperament perspective, divides the octave into 1600 equal parts of exactly 750 millicents each.

Theory

1600edo is a very strong 37-limit system, being distinctly consistent in the 37-limit with a smaller relative error than anything else with this property until 4501. It is also the first division past 311 with a lower 43-limit relative error.

In the 5-limit, it supports kwazy. In the 11-limit, it supports the rank-3 temperament thor. In higher limits, it tempers out 12376/12375 in the 17-limit and due to being consistent higher than 33-odd-limit it enables the essentially tempered flashmic chords.

Odd harmonics

Approximation of prime harmonics in 1600edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.045	-0.064	+0.174	-0.068	+0.222	+0.045	+0.237	+0.226	+0.173	+0.214
	Relative (%)	+0.0	+6.0	-8.5	+23.2	-9.1	+29.6	+5.9	+31.6	+30.1	+23.0	+28.6
Steps (reduced)		1600 (0)	2536 (936)	3715 (515)	4492 (1292)	5535 (735)	5921 (1121)	6540 (140)	6797 (397)	7238 (838)	7773 (1373)	7927 (1527)

Subsets and supersets

1600's divisors are 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, 800.

One step of it is the relative cent for 16. It's high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for interval size measure. One step of 1600edo is already used as a measure called śata in the context of 16edo Armodue theory.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3.5	[-53 10 16⟩, [26 -75 40⟩	[⟨1600 2536 3715]]	-0.0003	0.0228	3.04
2.3.5.7	4375/4374, [36 -5 0 -10⟩, [-17 5 16 -10⟩	[⟨1600 2536 3715 4492]]	-0.0157	0.0332	4.43
2.3.5.7.11	3025/3024, 4375/4374, [24 -1 -5 0 1⟩, [15 1 7 -8 -3⟩	[⟨1600 2536 3715 4492 5535]]	-0.0172	0.0329	4.39
2.3.5.7.11.13	3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543	[⟨1600 2536 3715 4492 5535 5921]]	-0.0087	0.0356	4.75
2.3.5.7.11.13.17	2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869	[⟨1600 2536 3715 4492 5535 5921 6540]]	-0.0163	0.0331	4.41

Rank-2 temperaments

Periods per 8ve	Generator	Cents	Associated Ratio	Temperaments
2	217\1600	162.75	1125/1024	Kwazy
32	23\1600	17.25	?	Dam / dike / polder
32	121\1600 (21/1600)	90.75 (15.75)	48828125/46294416 (?)	Windrose
32	357\1600 (7\1600)	267.75 (5.25)	245/143 (?)	Germanium
80	629\1600 (9\1600)	471.75 (6.75)	130/99 (?)	Tetraicosic